Chapter 6.3, Problem 28PPS

To determine

To prove:

The coordinate proof for the following:

‘If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a

parallelogram’

Explanation of Solution

Given information:

  $\begin{array}{l}\overline{\mathrm{A}\mathrm{B}}\cong \overline{\mathrm{C}\mathrm{D}}\\ \overline{\mathrm{A}\mathrm{D}}\cong \overline{\mathrm{B}\mathrm{C}}\end{array}$

Proof:

First draw a quadrilateral $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$

as below:

  

Now, the slope of $\overline{\mathrm{A}\mathrm{D}}$ is calculated as:

  $\overline{\mathrm{A}\mathrm{D}}=\frac{\mathrm{c}-0}{\mathrm{b}-0}=\frac{\mathrm{c}}{\mathrm{b}}$

The slope of $\overline{\mathrm{A}\mathrm{B}}$ is $0$ .

The slope of $\overline{\mathrm{B}\mathrm{C}}$ is calculated as:

  $\overline{\mathrm{B}\mathrm{C}}=\frac{\mathrm{c}-0}{\mathrm{b}+\mathrm{a}-\mathrm{a}}=\frac{\mathrm{c}}{\mathrm{b}}$

The slope of $\overline{\mathrm{C}\mathrm{D}}$ is $0$ .

Therefore, $\overline{\mathrm{A}\mathrm{D}}\parallel \overline{\mathrm{B}\mathrm{C}}$ and $\overline{\mathrm{A}\mathrm{B}}\parallel \overline{\mathrm{B}\mathrm{C}}$ .

Therefore, by definition of a parallelogram: $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is a parallelogram.

Hence proved